There are a few economic concepts that are widely cited (if not understood) by non-economists. Certainly, the “law” of supply and demand is one of them. The Keynesian (fiscal) multiplier is another; it is

the ratio of a change in national income to the change in government spending that causes it. More generally, the exogenous spending multiplier is the ratio of a change in national income to any autonomous change in spending (private investment spending, consumer spending, government spending, or spending by foreigners on the country’s exports) that causes it.

The multiplier is usually invoked by pundits and politicians who are anxious to boost government spending as a “cure” for economic downturns. What’s wrong with that? If government spends an extra $1 to employ previously unemployed resources, why won’t that $1 multiply and become $1.50, $1.60, or even $5 worth of additional output?

What’s wrong is the phony math by which the multiplier is derived, and the phony story that was long ago concocted to explain the operation of the multiplier.

**MULTIPLIER MATH**

To show why the math is phony, I’ll start with a derivation of the multiplier. The derivation begins with the accounting identity Y = C + I + G, which means that total output (Y) = consumption (C) + investment (I) + government spending (G). I could use a more complex identity that involves taxes, exports, and imports. But no matter; the bottom line remains the same, so I’ll keep it simple and use Y = C + I + G.

Keep in mind that the aggregates that I’m writing about here — Y , C , I , G, and later S — are supposed to represent real quantities of goods and services, not mere money. Keep in mind, also, that Y stands for gross domestic product (GDP); there is no real income unless there is output, that is, product.

Now for the derivation (right-click to enlarge this and later images):

So far, so good. Now, let’s say that b = 0.8. This means that income-earners, on average, will spend 80 percent of their additional income on consumption goods (C), while holding back (saving, S) 20 percent of their additional income. With b = 0.8, k = 1/(1 – 0.8) = 1/0.2 = 5. That is, every $1 of additional spending — let us say additional government spending (∆G) rather than investment spending (∆I) — will yield ∆Y = $5. In short, ∆Y = k(∆G), as a theoretical maximum. (Even if the multiplier were real, there are many things that would cause it to fall short of its theoretical maximum; see this, for example.)

How is it supposed to work? The initial stimulus (∆G) creates income (don’t ask how), a fraction of which (b) goes to C. That spending creates new income, a fraction of which goes to C. And so on. Thus the first round = ∆G, the second round = b(∆G), the third round = b(b)(∆G) , and so on. The sum of the “rounds” asymptotically approaches k(∆G). (What happens to S, the portion of income that isn’t spent? That’s part of the complicated phony story that I’ll examine in a future post.)

Note well, however, that the resulting ∆Y isn’t properly an increase in Y, which is an annual rate of output; rather, it’s the cumulative increase in total output over an indefinite number and duration of ever-smaller “rounds” of consumption spending.

The cumulative effect of a sustained increase in government spending might, after several years, yield a new Y — call it Y’ = Y + ∆Y. But it would do so only if ∆G persisted for several years. To put it another way, ∆Y persists only for as long as the effects of ∆G persist. The multiplier effect disappears after the “rounds” of spending that follow ∆G have played out.

The multiplier effect is therefore (at most) temporary; it vanishes after the withdrawal of the “stimulus” (∆G). The idea is that ∆Y should be temporary because a downturn will be followed by a recovery — weak or strong, later or sooner.

*An aside is in order here: Proponents of big government like to trumpet the supposedly stimulating effects of G on the economy when they propose programs that would lead to permanent increases in G, holding other things constant. And other things (other government programs) are constant (at least) because they have powerful patrons and constituents, and are harder to kill than Hydra. If the proponents of big government were aware of the economically debilitating effects of G and the things that accompany it (e.g., regulations), most of them would simply defend their favorite programs all the more fiercely.*

**WHY MULTIPLIER MATH IS PHONY MATH**

Now for my *exposé* of the phony math. I begin with Steven Landsburg, who borrows from the late Murray Rothbard:

. . . We start with an accounting identity, which nobody can deny:

Y = C + I + G. . . Since all output ends up somewhere, and since households, firms and government exhaust the possibilities, this equation must be true.

Next, we notice that people tend to spend, oh, say about 80 percent of their incomes. What they spend is equal to the value of what ends up in their households, which we’ve already called C. So we have

C = .8Y Now we use a little algebra to combine our two equations and quickly derive a new equation:

Y = 5(I+G) That 5 is the famous Keynesian multiplier. In this case, it tells you that if you increase government spending by one dollar, then economy-wide output (and hence economy-wide income) will increase by a whopping

fivedollars. What a deal!. . . [I]t was Murray Rothbard who observed that the really neat thing about this argument is that you can do exactly the same thing with any accounting identity. Let’s start with this one:

Y = L + E

Here Y is economy-wide income, L is Landsburg’s income, and E is everyone else’s income. No disputing that one.

Next we observe that everyone else’s share of the income tends to be about 99.999999% of the total. In symbols, we have:

E = .99999999 Y

Combine these two equations, do your algebra, and voila:

Y = 100,000,000 LThat 100,000,000 there is the soon-to-be-famous “Landsburg multiplier”. Our equation proves that if you send Landsburg a dollar, you’ll generate $100,000,000 worth of income for everyone else.

The policy implications are unmistakable. It’s just Eco 101!! [“The Landsburg Multiplier: How to Make Everyone Rich”,

The Big Questionsblog, June 25, 2013]

Landsburg attributes the nonsensical result to the assumption that

equations describing behavior would remain valid after a policy change. Lucas made the simple but pointed observation that this assumption is almost never justified.

. . . None of this means that you can’t write down [a] sensible Keynesian model with a multiplier; it does mean that the Eco 101 version of the Keynesian cross is not an example of such. This in turn calls into question the wisdom of the occasional pundit [Paul Krugman] who repeatedly admonishes us to be guided in our policy choices by the lessons of Eco 101. (“Multiple Comments”,

op. cit,, June 26, 2013)

It’s worse than that, as Landsburg almost acknowledges when he observes (correctly) that Y = C + I + G is an accounting identity. That is to say, it* isn’t* a functional representation — a model — of the dynamics of the economy. Assigning a value to b (the marginal propensity to consume) — even if it’s an empirical value — doesn’t alter that fact that the derivation is nothing more than the manipulation of a non-functional relationship, that is, an accounting identity.

Consider, for example, the equation for converting temperature Celsius (*C*) to temperature Fahrenheit (*F*): *F* = 32 + 1.8*C*. It follows that an increase of 10 degrees *C* implies an increase of 18 degrees *F*. This could be expressed as ∆*F/*∆*C** = k** , where *k** represents the “Celsius multiplier”. There is no mathematical difference between the derivation of the investment/government-spending multiplier (k) and the derivation of the Celsius multiplier (*k**). And yet we know that the Celsius multiplier is nothing more than a tautology; it tells us nothing about *how* the temperature rises by 10 degrees *C* or 18 degrees *F*. It simply tells us that when the temperature rises by 10 degrees *C*, the equivalent rise in temperature *F* is 18 degrees. The rise of 10 degrees *C* doesn’t *cause* the rise of 18 degrees *F*.

Similarly, the Keynesian investment/government-spending multiplier simply tells us that if ∆Y = $5 trillion, and if b = 0.8, then it is a matter of mathematical necessity that ∆C = $4 trillion and ∆I + ∆G = $1 trillion. In other words, a rise in I + G of $1 trillion *doesn’t caus*e a rise in Y of $5 trillion; rather, Y *must* rise by $5 trillion for C to rise by $4 trillion and I + G to rise by $1 trillion. If there’s a causal relationship between ∆G and ∆Y, the multiplier doesn’t portray it.

**PHONY MATH DOESN’T EVEN ADD UP
**

Recall the story that’s supposed to explain how the multiplier works: The initial stimulus (∆G) creates income, a fraction of which (b) goes to C. That spending creates new income, a fraction of which goes to C. And so on. Thus the first round = ∆G, the second round = b(∆G), the third round = b(b)(∆G) , and so on. The sum of the “rounds” asymptotically approaches k(∆G). So, if b = 0.8, k = 5, and ∆G = $1 trillion, the resulting cumulative ∆Y = $5 trillion (in the limit). And it’s all in addition to the output that would have been generated in the absence of ∆G, as long as many conditions are met. Chief among them is the condition that the additional output in each round is generated by resources that had been unemployed.

In addition to the fact that the math behind the multiplier is phony, as explained above, it also yields contradictory results. If one can derive an investment/government-spending multiplier, one can also derive a “consumption multiplier”:

Taking b = 0.8, as before, the resulting value of k_{c} is 1.25. Suppose the initial round of spending is generated by C instead of G. (I won’t bother with a story to explain it; you can easily imagine one involving underemployed factories and unemployed persons.) If ∆C = $1 trillion, shouldn’t cumulative ∆Y = $5 trillion? After all, there’s no essential difference between spending $1 trillion on a government project and $1 trillion on factory output, as long as both bursts of spending result in the employment of underemployed and unemployed resources (among other things).

But with k_{c} = 1.25, the initial $1 trillion burst of spending (in theory) results in additional output of only $1.25 trillion. Where’s the other $3.75 trillion? Nowhere. The $5 trillion is phony. What about the $1.25 trillion? It’s phony, too. The “consumption multiplier” of 1.25 is simply the inverse of b, where b = 0.8. In other words, Y must rise by $1.25 trillion if C is to rise by $1 trillion. More phony math.

**CAN AN INCREASE IN G HELP IN THE SHORT RUN?**

Can an exogenous increase in G spending really yield a short-term, temporary increase in GDP? Perhaps, but there’s many a slip between cup and lip. The following example goes beyond the bare theory of the Keynesian multiplier to address several practical and theoretical shortcomings (some which are discussed “here” and “here“):

- Annualized real GDP (Y) drops from $16.5 trillion a year to $14 trillion a year because of the unemployment of resources. (How that happens is a different subject.)
- Government spending (G) is temporarily and quickly increased by an annual rate of $500 billion; that is, ∆G = $0.5 trillion. The idea is to restore Y to $16 trillion, given a multiplier of 5 (In standard multiplier math: ∆Y = (k)(∆G), where k = 1/(1 – MPC); k = 5, where MPC = 0.8.)
- The ∆G is financed in a way that doesn’t reduce private-sector spending. (This is almost impossible, given Ricardian equivalence — the tendency of private actors to take into account the long-term, crowding-out effects of government spending as they make their own spending decisions. The closest approximation to neutrality can be attained by financing additional G through money creation, rather than additional taxes or borrowing that crowds out the financing of private-sector consumption and investment spending.)
- To have the greatest leverage, ∆G must be directed so that it employs only those resources that are idle, which then acquire purchasing power that they didn’t have before. (This, too, is almost impossible, given the clumsiness of government.)
- A fraction of the new purchasing power flows, through consumption spending (C), to the employment of other idle resources. That fraction is called the marginal propensity to consume (MPC), which is the rate at which the owners of idle resources spend additional income on so-called consumption goods. (As many economists have pointed out, the effect could also occur as a result of investment spending. A dollar spent is a dollar spent, and investment spending has the advantage of directly enabling economic growth, unlike consumption spending.)
- A remainder goes to saving (S) and is therefore available for investment (I) in future production capacity. But S and I are ignored in the multiplier equation: One story goes like this: S doesn’t elicit I because savers hoard cash and investment is discouraged by the bleak economic outlook. Here is a more likely story: The multiplier would be infinite (and therefore embarrassingly inexplicable) if S generated an equivalent amount of I, because the marginal propensity to spend (MPS) would be equal to 1, and the multiplier equation would look like this: k = 1/(1 – MPS) = ∞, where MPS = 1.
- In any event, the initial increment of C (∆C) brings forth a new “round” of production, which yields another increment of C, and so on,
*ad infinitum*. If MPC = 0.8, then assuming away “leakage” to taxes and imports, the multiplier = k = 1/(1 – MPC), or k = 5 in this example. (The multiplier rises with MPC and reaches infinity if MPC = 1. This suggests that a very high MPC is economically beneficial, even though a very high MPC implies a very low rate of saving and therefore a very low rate of growth-producing investment.) - Given k = 5, ∆G = $0.5T would cause an
increase in real output of $2.5 trillion (assuming no “leakage” or offsetting reductions in private consumption and investment); that is, ∆Y = [k][∆G]= $2.5 trillion. However, because G and Y usually refer to annual rates, this result is mathematically incoherent; ∆G = $0.5 trillion does not restore Y to $16.5 trillion.*eventual* - In any event, the increase in Y isn’t permanent; the multiplier effect disappears after the “rounds” resulting from ∆G have played out. If the theoretical multiplier is 5, and if transactional velocity is 4 (i.e., 4 “rounds” of spending in a year), more than half of the multiplier effect would be felt within a year from each injection of spending, and about two-thirds would be felt within two years of each injection. It seems unlikely, however, that the multiplier effect would be felt for much longer, because of changing conditions (e.g., an exogenous boost in private investment, private reemployment of resources, discouraged workers leaving the labor force, shifts in expectations about inflation and returns on investment).
- All of this ignores that fact that the likely cause of the drop in Y is not insufficient “aggregate demand”, but a “credit crunch” (Michael D. Bordo and Joseph G. Haubrich in “Credit Crises, Money, and Contractions: A Historical View,” Federal Reserve Bank of Cleveland, Working Paper 09-08, September 2009). “Aggregate demand” doesn’t exist, except as an after-the-fact measurement of the money value of goods and services comprised in Y. “Aggregate demand”, in other words, is merely the sum of millions of individual transactions, the rate and total money value of which decline for specific reasons, “credit crunch” being chief among them. Given that, an exogenous increase in G is likely to yield a real increase in Y only if the increase in G leads to an increase in the money supply (as it is bound to do when the Fed, in effect, prints money to finance it). But because of cash hoarding and a bleak investment outlook, the increase in the money supply is unlikely to generate much additional economic activity.

So much for that.

**THE THEORETICAL MAXIMUM**

A somewhat more realistic version of multiplier math — as opposed to the version addressed earlier — yields a maximum value of k = 1:

How did I do that? In step 3, I made C a function of P (private-sector GDP) instead of Y (usually taken as the independent variable). Why? C is more closely linked to P than to Y, as an analysis of GDP statistics will prove. (Go here, download the statistics for the post-World War II era from tables 1.1.5 and 3.1, and see for yourself.)

**THE TRUE MULTIPLIER**

In fact, a sustained increase in government spending will have a negative effect on real output — a multiplier of less than 1, in other words.

Robert J. Barro of Harvard University opens an article in *The Wall Street Journal* with the statement that “economists have not come up with explanations … for multipliers above one”. Barro continues:

A much more plausible starting point is a multiplier of zero. In this case, the GDP is given, and a rise in government purchases requires an equal fall in the total of other parts of GDP — consumption, investment and net exports….

What do the data show about multipliers? Because it is not easy to separate movements in government purchases from overall business fluctuations, the best evidence comes from large changes in military purchases that are driven by shifts in war and peace. A particularly good experiment is the massive expansion of U.S. defense expenditures during World War II. The usual Keynesian view is that the World War II fiscal expansion provided the stimulus that finally got us out of the Great Depression. Thus, I think that most macroeconomists would regard this case as a fair one for seeing whether a large multiplier ever exists.

I have estimated that World War II raised U.S. defense expenditures by $540 billion (1996 dollars) per year at the peak in 1943-44, amounting to 44% of real GDP. I also estimated that the war raised real GDP by $430 billion per year in 1943-44. Thus, the multiplier was 0.8 (430/540). The other way to put this is that the war lowered components of GDP aside from military purchases. The main declines were in private investment, nonmilitary parts of government purchases, and net exports — personal consumer expenditure changed little. Wartime production siphoned off resources from other economic uses — there was a dampener, rather than a multiplier….

There are reasons to believe that the war-based multiplier of 0.8 substantially overstates the multiplier that applies to peacetime government purchases. For one thing, people would expect the added wartime outlays to be partly temporary (so that consumer demand would not fall a lot). Second, the use of the military draft in wartime has a direct, coercive effect on total employment. Finally, the U.S. economy was already growing rapidly after 1933 (aside from the 1938 recession), and it is probably unfair to ascribe all of the rapid GDP growth from 1941 to 1945 to the added military outlays. [“Government Spending Is No Free Lunch”,

The Wall Street Journal, January 22, 2009]

This is from a paper by Valerie A. Ramsey:

… [I]t appears that a rise in government spending does not stimulate private spending; most estimates suggest that it significantly lowers private spending. These results imply that the government spending multiplier is below unity. Adjusting the implied multiplier for increases in tax rates has only a small effect. The results imply a multiplier on total GDP of around 0.5. [“Government Spending and Private Activity”, National Bureau of Economic Research, January 2012]

There is a key component of government spending which usually isn’t captured in estimates of the multiplier: transfer payments, which are mainly “social benefits” (e.g., Social Security, Medicare, and Medicaid). In fact, actual government spending in the U.S., including transfer payments, is about double the nominal amount that is represented in G, the standard measure of government spending (the actual cost of government operations, buildings, equipment, etc.). But transfer payments — like other government spending — are subsidized by directing resources from persons who are directly productive (active worker) and whose investments are directly productive (innovators, entrepreneurs, stockholders, etc.) to persons who (for the most part) are economically unproductive and counterproductive. It follows that real economic output must be affected by transfer payments.

Other factors are also important to economic growth, namely, private investment in business assets, the rate at which regulations are being issued, and inflation. The combined effects of these factors and aggregate government spending have been estimated:. I borrow from that estimate, with a slight, immaterial change in nomenclature:

g_{r}= 0.0275 -0.347F + 0.0769A – 0.000327R – 0.135P

Where,

g= real rate of GDP growth in a 10-year span (annualized)_{r}

F= fraction of GDP spent by governments at all levels during the preceding 10 years [including transfer payments]

A= the constant-dollar value of private nonresidential assets (business assets) as a fraction of GDP, averaged over the preceding 10 years

R= average number of Federal Register pages, in thousands, for the preceding 10-year period

P= growth in the CPI-U during the preceding 10 years (annualized).The r-squared of the equation is 0.73 and the F-value is 2.00E-12. The p-values of the intercept and coefficients are 0.099, 1.75E-07, 1.96E-08, 8.24E-05, and 0.0096. The standard error of the estimate is 0.0051, that is, about half a percentage point.

Assume, for the sake of argument, that F rises while the other independent variables remain unchanged. A rise in F from 0.24 to 0.33 (the actual change from 1947 to 2007) would reduce the real rate of economic growth by 0.031 percentage points. The real rate of growth from 1947 to 1957 was 4 percent. Other things being the same, the rate of growth would have dropped to 0.9 percent in the period 2008-2017. It actually dropped to 1.4 percent, which is within the standard error of the equation. And the discrepancy could be the result of changes in the other variables — a disproportionate increase in business assets (A), for example.

Given that r_{g} = -0.347F, other things being the same, then_{
}

Y_{1} = Y_{0}(c – 0.347F)

Where,

Y_{1} = real GDP in the period after a change in F, other things being the same

Y_{0} = real GDP in the period during which F changes

c = a constant, representing the sum of 1 + 0.025 + the coefficients obtained from fixed values of A, R, and P

The true F multiplier, k_{T}, is therefore negative:

k_{T} = ∆Y/∆F = -0.347Y_{0}

For example, with Y_{0 }= 1000 , F =0 , and other things being the same,

∆Y = [1000 – (o)(1000)] = 1000, when F = 0

∆Y = [1000 – (-0.347)(1000)] = 653, when F = 1

Keeping in mind that the equation is based on an analysis of successive 10-year periods, the true F multiplier should be thought of as representing the effect of a change in the average value of F in a 10-year period on the average value of Y in a subsequent 10-year period.

This is not to minimize the deleterious effect of F (and other government-related factors) on Y. If the 1947-1957 rate of growth (4 percent) had been sustained through 2017, Y would have risen from $1.9 trillion in 1957 to $20 trillion in 2017. But because F, R, and P rose markedly over the years, the real rate of growth dropped sharply and Y reached only $17.1 trillion in 2017. That’s a difference of almost $3 trillion in a single year.

Such losses, summed over several decades, represent millions of jobs that weren’t created, significantly lower standards of living, greater burdens on the workers who support retirees and subsidize their medical care, and the loss of liberty that inevitably results when citizens are subjugated to tax collectors and regulators.